A straight line intersects sides AB, BC and the extension of side AC of a
triangle ABC at points D, E and F respectively. Prove that the midpoints
of the line segments DC, AE and BF lies on a straight line.

In triangle ABC, AB = AC. A circle is tangent internally to the
circumcircle of triangle ABC and also to sides AB and AC at points P and
Q, respectively. How can I prove that the midpoint of the segment PQ is
the center of the mixtilinear incircle of triangle ABC?

Given right triangles ABC and DCB with rt angles at B and C, triangle
ABC's hypotenuse 20 and triangle DCB's hypotenuse 30. The hypotenuses
intersect at point E, a distance of 10 from BC. Find the length of BC.